Kramers–Kronig relations

The Kramers–Kronig relations are bidirectional mathematical relations, connecting the real and imaginary parts of any complex function that is analytic in the upper half-plane. The relations are often used to compute the real part from the imaginary part (or vice versa) of response functions in physical systems, because for stable systems, causality implies the condition of analycity, and conversely, analyticity implies causality of the corresponding stable physical system. The relation is named in honor of Ralph Kronig and Hans Kramers. In mathematics, these relations are known by the names Sokhotski–Plemelj theorem and Hilbert transform. Let be a complex function of the complex variable , where and are real. Suppose this function is analytic in the closed upper half-plane of and diminishes faster than as . Slightly weaker conditions are also possible. The Kramers–Kronig relations are given by and where is real and where denotes the Cauchy principal value. The real and imaginary parts of such a function are not independent, allowing the full function to be reconstructed given just one of its parts. The proof begins with an application of Cauchy's residue theorem for complex integration. Given any analytic function in the closed upper half-plane, the function , where is real, is analytic in the (open) upper half-plane. The residue theorem consequently states that for any closed contour within this region. When the contour is chosen to trace the real axis, a hump over the pole at , and a large semicircle in the upper half-plane. This follows decomposition of the integral into its contributions along each of these three contour segments and pass them to limits. The length of the semicircular segment increases proportionally to , but the integral over it vanishes in the limit because vanishes faster than . We are left with the segments along the real axis and the half-circle around the pole. We pass the size of the half-circle to zero and obtain The second term in the last expression is obtained using the theory of residues, more specifically, the Sokhotski–Plemelj theorem.